I have faced some problem in computing surface integral of some vectors in spherical polar coordinate system I have studied that in genaral infinitesimal surface differential vector has three component along $r$, $θ$ and $\phi$ cap direction
$$ds=r^2\sinθ\mathrm dθ\mathrm d\phi r' + r\sinθ\mathrm dr\mathrm d\phi θ' + r\mathrm dr\mathrm dθ\phi'$$ Where $r',\phi',θ'$ are the unit vectors So suppose I have a vector $$ v=Ar'+bθ' $$ And I want to calculate
$$∮v.\mathrm ds.$$ over a spherical surface

Then on which surface components do I have to integrate ?? I mean on spherical surface only there radius is constant so $\mathrm dr=0$ ! But my problem is then what are actually the other surface components are here? I mean if it be a solid sphere then it might have all three surface differential components then how can I understand all surface differential components, only the surface i understand is the normal to the outer surface which is along the $r'$.

Can you show me the other components with picture?

  • $\begingroup$ Might Mathematics be better suited for your math question? $\endgroup$
    – Kyle Kanos
    May 28 '17 at 12:51

You need to parametrize the surface you are trying to integrate over. Then you need its normal vector in the base of your spherical unit vectors. If you have parametrized correctly, you should already know the interval for each coordinate, it is part of what describes the surface you are trying to integrate over.

Once you have all that, then just dot product both vectors, and it becomes a standard triple or double integral over a set region, depending on the surface.


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